# Symmetric sum

The symmetric sum $\sum_{\rm sym} f(x_1, x_2, x_3, \dots, x_n)$ of a function $f(x_1, x_2, x_3, \dots, x_n)$ of $n$ variables is defined to be $\sum_{\sigma} f(x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}, \dots, x_{\sigma(n)})$, where $\sigma$ ranges over all permutations of $(1, 2, 3, \dots, n)$.

More generally, a symmetric sum of $n$ variables is a sum that is unchanged by any permutation of its variables.

Any symmetric sum can be written as a polynomial of elementary symmetric sums.

A symmetric function of $n$ variables is a function that is unchanged by any permutation of its variables. The symmetric sum of a symmetric function $f(x_1, x_2, x_3, \dots, x_n)$ therefore satisfies $$\sum_{\rm sym} f(x_1, x_2, x_3, \dots, x_n) = n!f(x_1, x_2, x_3, \dots, x_n).$$

Given $n$ variables $x_1,\ldots,x_n$ and a symmetric function $f(x_1,\ldots,x_r)$ with $r\leq n$, the notation $\sum_{\rm sym}f(x_1, x_2, x_3, \dots, x_r)$ is sometimes used to denote the sum of $f(x_1,\ldots,x_r)$ over all $\left(\begin{matrix}n\cr r\end{matrix}\right)$ subsets of size $r$ in $\{x_1,\ldots,x_n\}$.